T1 separation axiom
E898480
The T1 separation axiom is a topological property requiring that for any two distinct points, each has an open set containing it but not the other, ensuring all singletons are closed.
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
separation axiom
ⓘ
topological property ⓘ |
| alsoKnownAs |
Fréchet axiom
NERFINISHED
ⓘ
T1 axiom NERFINISHED ⓘ |
| appliesTo | topological space ⓘ |
| category | mathematics ⓘ |
| characterizes | spaces in which convergence of sequences is determined by limits being unique when the space is also Hausdorff ⓘ |
| closureProperty | The closure of a singleton {x} is {x} itself. ⓘ |
| consequence | No point is isolated by being contained in every nonempty open set unless the space is trivial ⓘ |
| definition | For any two distinct points x and y in a topological space, there exists an open set containing x but not y, and an open set containing y but not x. ⓘ |
| ensures | points are topologically distinguishable by closed sets ⓘ |
| equivalentCondition |
Every singleton set {x} is closed in the space.
ⓘ
Finite subsets of the space are closed. ⓘ For each point x and each point y ≠ x, there exists an open set containing x but not y. ⓘ |
| failsIn | indiscrete topology on a set with more than one point ⓘ |
| field | topology ⓘ |
| formalizes | the idea that points can be separated by open sets in at least one direction ⓘ |
| historicalNote | Named after Maurice Fréchet in some literature ⓘ |
| holdsIn |
discrete topology on any set
ⓘ
standard topology on the real numbers ⓘ |
| implies | T0 separation axiom ⓘ |
| interactionWithCompactness | In a T1 space, compact subsets are closed ⓘ |
| interactionWithConnectedness | In a T1 space, components are intersections of clopen sets ⓘ |
| interactionWithConvergence | In a T1 space, a sequence can converge to a point only if every neighborhood of the point contains all but finitely many terms of the sequence ⓘ |
| isImpliedBy |
Hausdorff property
NERFINISHED
ⓘ
T2 separation axiom ⓘ normal Hausdorff property ⓘ regular Hausdorff property ⓘ |
| isStrongerThan | T0 separation axiom NERFINISHED ⓘ |
| isWeakerThan |
Hausdorff property
ⓘ
T2 separation axiom ⓘ |
| logicalStrength | strictly stronger than T0 and strictly weaker than T2 in general ⓘ |
| notPreservedUnder | arbitrary quotient maps ⓘ |
| preservedUnder |
arbitrary products
ⓘ
continuous open images ⓘ finite products ⓘ taking subspaces ⓘ |
| topologicalInvariance | invariant under homeomorphisms ⓘ |
| usedIn |
general topology
ⓘ
topological separation theory ⓘ |
| usedToDefine | T1 space ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Hausdorff space